1. Problem Solving with Algorithms
and Data Structure — Graph
Bruce Tsai

2. http://interactivepython.org/runestone/
static/pythonds/Graphs/Objectives.html

3. Graph
A set of objects where some pairs of objects are
connected by links
road map
airline flights
internet connection
skill tree
Tree is special kind of graph
Vertex
(node)
Edge
(arc)
3

4. Goal
Represent problem in form of graph
Use graph algorithms to solve problem
4

5. (Undirected) Graph
G = (V, E)
V = {v_1, v_2, v_3, v_4, v_5, v_6}
E = {(v_1, v_2), (v_1, v_5), (v_2, v_3), (v_2,
v_5), (v_3, v_4), (v_4, v_5), (v_4, v_6)}
(v_1, v_5) == (v_5, v_1)
5

6. Directed Graph
D = (V, A)
V = {v_0, v_1, v_2, v_3, v_4, v_5}
A = {(v_0, v_1), (v_0, v_5), (v_1, v_2), (v_2,
v_3), (v_3, v_5), (v_3, v_4), (v_4, v_0), (v_5,
v_2), (v_5, v_4)}
6

7. Attributed Graph
Attribute
Characters of graph vertices or edges
weight, color, anything you want
7

8. Path and Cycle
Path
Sequence of vertices that are connected by edges
[v_1, v_5, v_4, v_6]
Cycle
A path that starts and ends at same vertex
[v_2, v_3, v_4, v_5, v_2]
8

9. Adjacency Matrix
9
from
to

10. Discussion Question
A
B
C
D
E
F
7
5
1
2
7
3
2
8
1
2
4
5
6
1
8
10

11. Adjacency List
11

12. Discussion Question
1
2
3
4
5
6
10
15
5
7
7
10
7
13
5
12

13. Word Ladder Problem
Make change occur gradually by changing
one letter at a time
FOOL -> POOL -> POLL -> POLE -> PALE ->
SALE -> SAGE
13

14. Word Ladder Graph
14

16. O(V*L)
O(2E)
O(V+E)

18. Breadth First Search (BFS)
18

19. BFS Analysis
O(V+E)
19

20. Discussion Question - BFS
[1]
[1, 2, 3, 6]
[2, 3, 6]
[3, 6, 4]
[6, 4, 5]
[4, 5]
[5]
1
2
3
4
5
6
20

21. Knight’s Tour Problem
21

22. Legal Move Graph
22

24. Knight Graph
24

27. Knight’s Tour
27

28. Choose Better Next Vertex
Visit hard-to-reach corners early
Use the middle square to hop across board
only when necessary
28

30. General Depth First Search (DFS)
Knight’s Tour is special case of depth first
search
create the depth first tree
General depth first search is to search as
deeply as possible
30

31. O(V+E)
O(V)

32. queue
stack

33. Computational Complexity

34. P versus NP problem
P: questions for which some algorithms can
provide an answer in polynomial time
NP: questions for which an answer can be
verified in polynomial time
Subset sum problem
{-7, -3, -2, 5, 8} -> {-3, -2, 5}
34

35. decision problem set
optimization problems
search problems

36. NP-complete
Reduction: algorithm for transforming one
problem into another problem
NP-complete: a problem p in NP is NP-
Complete if every other problem in NP can
be transformed into p in polynomial time
hardest problems in NP
36

37. NP-hard
Decision problem: any arbitrary yes-or-no
question on an infinite set of inputs
NP-hard: a decision problem H in NP-hard
when for any problem L in NP, there is
polynomial-time reduction from L to H
at least as hard as the hardest problems in
NP
37

38. Topological Sorting
Linear ordering of all vertices of a directed
graph
(u, v) is an edge of directed graph and u is
before v in ordering
38

40. Start and Finish Time

41. Implementation
1. Call dfs(g)
2. Store vertices based on decreasing order of
finish time
3. Return the ordered list
41

42. Lemma
A directed graph G is acyclic if and only if a
depth-first search of G yields no back edges.
Back edge: edge (u, v) connecting vertex u to
an ancestor v in depth-first tree (v ↝ u → v)
=>: back edge exists then cycle exists
<=: cycle exists then back edge exists
42

43. Proof of Implementation
For any pair of distinct vertices u, v ∈ V, if there is an
edge in G from u to v, then f[v] < f[u].
Consider edge (u, v) explored by DFS, v cannot be
gray since then that edge would be back edge.
Therefore v must be white or black.
If v is white, v is descendant of u, and so f[v] < f[u]
If v is black, it has already been finished, so that
f[v] < f[u]
43

44. Strongly Connected Components
G = (V, E) and C ⊂ V
such that ∀ (v_i, v_j) ∈ C, path v_i to v_j
and path v_j to v_i both exist
C is strongly connected components (SCC)
44

46. Implementation
1. Call dfs(g)
2. Compute transpose of g as g_t
3. Call dfs(g_t) but explore each vertex in
decreasing order of finish time
4. Each tree of step 3 is a SCC
original transpose
46

47. dfs(g)

48. dfs(g_t)

49. Lemma
Let C and C’ be distinct SCCs in directed
graph G = (V, E), let u, v ∈ C, let u’, v’ ∈ C’,
and suppose that there is a path u ↝ u’ in G.
Then there cannot also be a path v’ ↝ v in G.
if v’ ↝ v exists then both u ↝ u’ ↝ v’ and v’
↝ v ↝ u exist
C and C’ are not distinct SCCs
49

50. Lemma
Let C and C’ be distinct SCCs in directed
graph G = (V, E). Suppose that there is an
edge (u, v) ∈ E, where u ∈ C and v ∈ C’. Then
f(C) > f(C’)
from x ∈ C to w ∈ C’
from y ∈ C’ cannot reach any vertex in C
50

51. Corollary
Let C and C’ be distinct SCCs in directed
graph G = (V, E). Suppose that there is an
edge (u, v) ∈ ET, where u ∈ C and v ∈ C’.
Then f(C) < f(C’).
(v, u) ∈ E then f(C’) < f(C)
51

52. Proof of Implementation

53. Shortest Path Problem
Find the path with the smallest total weight
along which to route any given message
53

54. Dijkstra’s Algorithm
Iterative algorithm providing the shortest
path from one particular starting node to all
other nodes in graph
Path distance, priority queue
54

59. O(V)
O(logV)
O(logV)
O(E*logV)
O(V*logV)
O((V+E)*logV)

60. Broadcast Problem

61. Minimum Spanning Tree
T is acyclic subset of E that connects all
vertices in V
The sum of weight of edges in T is minized
61

62. Prim’s Spanning Tree Algorithm
Special case of generic minimum spanning
tree
Find shortest paths in graph
62

65. Kruskal’s Algorithm
Sort edges in E into ascending order by
weight
For (u, v) ∈ E, if set of u not equal to set of v
A ← A ∪ {(u, v)}
65

68. Reference
http://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/
AproxAlgor/TSP/tsp.htm
http://en.wikipedia.org/wiki/Graph_%28mathematics%29
http://en.wikipedia.org/wiki/P_versus_NP_problem
http://en.wikipedia.org/wiki/NP-complete
http://en.wikipedia.org/wiki/NP-hard
http://en.wikipedia.org/wiki/Reduction_(complexity)
http://en.wikipedia.org/wiki/Knight%27s_tour
http://staff.ustc.edu.cn/~csli/graduate/algorithms/book6/chap24.htm
Introduction to Algorithms, 2nd edition